Moshe consummates his task as the “teacher of all Israel” by transmitting the completed Torah to his people:
And Moshe wrote this Torah, and he gave it to the Kohanim, sons of Levi, who carry the Ark of the Covenant of Hashem, and to all the elders of Israel (Devarim 31:9).

In the future, the Children of Israel will be in danger of straying from the path of the Torah. To remind them, Hashem commands that the song Ha’azinu (chapter 32) be written:
And now, write for yourselves this song (KITVU LACHEM ET HA’SHIRA HA’ZOT) and teach it to the Children of Israel, place it in their mouths, so that this song shall be for Me as a witness for the Children of Israel (verse 19).
But when Moshe fulfills this command, it is not only Ha’azinu that he writes:
And it was when Moshe completed writing the words of this Torah in a book in their entirety (AD TUMAM), that Moshe commanded the Levites, those who carry the Ark of Hashem’s Covenant, saying, “Take this book of the Torah and place it alongside the Ark of Hashem your G-d’s Covenant. And it shall be there as a witness for you” (verses 24-26).

He proceeds to teach the people as Hashem commanded, but again the text reverts to Ha’azinu:
And Moshe spoke in the ears of the whole congregation of Israel the words of this song in their entirety (AD TUMAM) (verse 30).

For 40 years, Moshe wrote down the Torah one section at a time (see Gittin 60b). Now when Hashem says, “write for yourselves this song”, it is tantamount to “Finish the Torah,” because the song is the finishing touch. The terms this song and this Torah are interchangeable; by writing the end of the Torah, Moshe has in effect written the whole Torah. [The instruction write for yourselves this song is thus an example of a self-referential statement, a concept which Douglas R. Hofstadter explores in Metamagical Themas.]

Our Sages understood that KITVU LACHEM, though in the plural, is addressed to each individual:
Rabba said, “Even though one’s fathers bequeathed him a Torah scroll, it is a commandment for him to write one of his own, as it says, ‘And now, write for yourselves this song’” (Sanhedrin 21b; see also Nedarim 38a).

This is the 613th mitzvah (Rambam, Book of the Commandments, Positive Commandment § 18; Sefer HaChinuch, ascribed to either R. Aharon HaLevi or R. Pinchas HaLevi of Barcelona, mid-13th Century). Rambam (“Laws of Writing Tefillin, Mezuzah and Sefer Torah”, 7:1) explains:
“There is a positive commandment for each and every man in Israel to write a Torah scroll for himself, as it says, “And now, write for yourselves this song.” This means to say, “Write for yourselves a Torah which contains this song,” because one does not write the Torah in separate sections. And even if his ancestors left him a Torah scroll it is a commandment to write his own. If he wrote it with his own hand, it is as if he received it at Mount Sinai; but if he does not know how to write, others write for him. Anyone who corrects a Torah scroll -- even one letter – it is as if he wrote all of it.”

Torah Temimah (R. Baruch ben Yechiel Michel HaLevi Epstein, 1860-1942) insists that the mitzvah to write one’s own Torah scroll is not derived from LACHEM, for yourselves – implying a duty to own (as in the case of the four species on Sukkot) – for then there should be no problem with inheritance. Rather, the source is KITVU, write. The purpose of this mitzvah is to increase the number of Torah scrolls in existence, thereby disseminating Torah study. This goal could not be accomplished by merely passing the same scrolls down from generation to generation.

Rav Joseph B. Soloveichik (1903-1993) in Shiurim leZecher Abba Mari, z”l (vol. I, p. 240 ff.) posits two concepts in connection with the Written Torah:
1) An item (cheftza) of Written Torah.
2) The sanctity of a Torah scroll.
The first derives from the verse
And Hashem said to Moshe, “Write for yourself these words” (Shemot 34:27).
At that point, the Torah was not yet complete, so it could not have referred to the creation of a whole Torah scroll. But, it did establish certain laws of the Written Torah, such as:
a) “Matters that are in writing [i.e., citations from the Written Torah] may not be said by heart” (Gittin 60b).
b) The requirement to score straight lines before writing a Biblical quote (see Gittin 7a).

The second concept derives from:
And Moshe wrote this Torah, … And it was when Moshe completed writing the words of this Torah in a book in their entirety (31: 9,24).
An incomplete Torah scroll is an inherent contradiction — it is invalid and lacks the sanctity of a sefer Torah. Consequently, one does not fulfill the 613th mitzvah therewith, nor do we read from it in public or use it in court to adjure one who must swear.

The words of the sefer Torah are written AD TUMAM (literally, “until their completing”), from the verb root T-M-M. Malbim (R. Meir Leib ben Yechiel Michael, 1809-1877) in his Biblical dictionary, Yair Or differentiates between K-L-H (stopping, even though more can be done) and T-M-M (completing, because no more can be done):
Finished (KALLU) are the prayers of David son of Yishai (Tehillim 72:20);
David has stopped composing prayers, but the praises of Hashem are never complete. However.
…completed (TAMU) are the words of Iyov (Iyov 31:40)
because there is nothing more to add.

The Torah which began with the world’s creation, concluded in terms of K-L-H:
And the heavens and the earth and all their host were finished (VAYECHULU) (Bereishit 2:1).
Creation ended but is not complete, because man must perfect it.

The Torah, on the other hand, is written AD TUMAM. It is perfect in itself, and must continue be replicated and taught forever.